from __future__ import annotations




def 	_UpperCamelCase (      snake_case__,     snake_case__ = None,     snake_case__ = None,     snake_case__ = False,     )    -> Union[str, Any]:
							__UpperCAmelCase    :      List[str]       = cipher_alphabet or [chr(a_				) for i in range(97,     123				)]

							# If the argument is None or the user provided an empty dictionary
							if not frequencies_dict:
														# Frequencies of letters in the english language (how much they show up)
														__UpperCAmelCase    :      Tuple       = {
														    "a": 0.0_8497,
														    "b": 0.0_1492,
														    "c": 0.0_2202,
														    "d": 0.0_4253,
														    "e": 0.1_1162,
														    "f": 0.0_2228,
														    "g": 0.0_2015,
														    "h": 0.0_6094,
														    "i": 0.0_7546,
														    "j": 0.0_0153,
														    "k": 0.0_1292,
														    "l": 0.0_4025,
														    "m": 0.0_2406,
														    "n": 0.0_6749,
														    "o": 0.0_7507,
														    "p": 0.0_1929,
														    "q": 0.0_0095,
														    "r": 0.0_7587,
														    "s": 0.0_6327,
														    "t": 0.0_9356,
														    "u": 0.0_2758,
														    "v": 0.0_0978,
														    "w": 0.0_2560,
														    "x": 0.0_0150,
														    "y": 0.0_1994,
														    "z": 0.0_0077,
														}
							else:
														# Custom frequencies dictionary
														__UpperCAmelCase    :      Dict       = frequencies_dict

							if not case_sensitive:
														__UpperCAmelCase    :      Dict       = ciphertext.lower()

							# Chi squared statistic values
							__UpperCAmelCase    :      Optional[Any]       = {}

							# cycle through all of the shifts
							for shift in range(len(a_				)				):
														__UpperCAmelCase    :      List[str]       = ""

														# decrypt the message with the shift
														for letter in ciphertext:
																					try:
																												# Try to index the letter in the alphabet
																												__UpperCAmelCase    :      Dict       = (alphabet_letters.index(letter.lower()				) - shift) % len(
																												    a_				)
																												decrypted_with_shift += (
																												    alphabet_letters[new_key].upper()
																												    if case_sensitive and letter.isupper()
																												    else alphabet_letters[new_key]
																												)
																					except ValueError:
																												# Append the character if it isn't in the alphabet
																												decrypted_with_shift += letter

														__UpperCAmelCase    :      Tuple       = 0.0

														# Loop through each letter in the decoded message with the shift
														for letter in decrypted_with_shift:
																					if case_sensitive:
																												__UpperCAmelCase    :      Tuple       = letter.lower()
																												if letter in frequencies:
																																			# Get the amount of times the letter occurs in the message
																																			__UpperCAmelCase    :      int       = decrypted_with_shift.lower().count(a_				)

																																			# Get the excepcted amount of times the letter should appear based
																																			# on letter frequencies
																																			__UpperCAmelCase    :      Union[str, Any]       = frequencies[letter] * occurrences

																																			# Complete the chi squared statistic formula
																																			__UpperCAmelCase    :      Tuple       = ((occurrences - expected) ** 2) / expected

																																			# Add the margin of error to the total chi squared statistic
																																			chi_squared_statistic += chi_letter_value
																					else:
																												if letter.lower() in frequencies:
																																			# Get the amount of times the letter occurs in the message
																																			__UpperCAmelCase    :      Tuple       = decrypted_with_shift.count(a_				)

																																			# Get the excepcted amount of times the letter should appear based
																																			# on letter frequencies
																																			__UpperCAmelCase    :      Optional[int]       = frequencies[letter] * occurrences

																																			# Complete the chi squared statistic formula
																																			__UpperCAmelCase    :      Union[str, Any]       = ((occurrences - expected) ** 2) / expected

																																			# Add the margin of error to the total chi squared statistic
																																			chi_squared_statistic += chi_letter_value

																												# Add the data to the chi_squared_statistic_values dictionary
														__UpperCAmelCase    :      Optional[int]       = (
														    chi_squared_statistic,
														    decrypted_with_shift,
														)

							# Get the most likely cipher by finding the cipher with the smallest chi squared
							# statistic
							def chi_squared_statistic_values_sorting_key(snake_case__				) -> tuple[float, str]:
														return chi_squared_statistic_values[key]

							__UpperCAmelCase    :      List[Any]       = min(
							    a_,     key=a_,     )

							# Get all the data from the most likely cipher (key, decoded message)
							(
							    (
							    __UpperCAmelCase							
							)     ,						(
							    __UpperCAmelCase							
							)     ,													
							)    :      Union[str, Any]       = chi_squared_statistic_values[most_likely_cipher]

							# Return the data on the most likely shift
							return (
							    most_likely_cipher,
							    most_likely_cipher_chi_squared_value,
							    decoded_most_likely_cipher,
							)










'''simple docstring'''


import math


def     SCREAMING_SNAKE_CASE							(   a_						:				float					,       a_						:				float      ):

  if initial_intensity < 0:
    raise ValueError('The value of intensity cannot be negative'      )
    # handling of negative values of initial intensity
  if angle < 0 or angle > 360:
    raise ValueError('In Malus Law, the angle is in the range 0-360 degrees'      )
    # handling of values out of allowed range
  return initial_intensity * (math.cos(math.radians(a_      )      ) ** 2)


if __name__ == "__main__":
  import doctest

  doctest.testmod(name="malus_law")



